1. Geometric Monte Carlo and Black Janus Geometries
Based on
IBS-KIAS
Joint Workshop on particle physics and Cosmology
5-10 February 2017, High1 Resort, Jeongsun, Korea
Kyung Kiu Kim (Yonsei University)
With Dongsu Bak(University of Seoul), Chanju Kim(Ewha Univ.), Hyunsoo Min(University of Seoul) and Jeong-Pil Song(Brown Univ.)

2. Contents Introduction 3D Black Janus deformation
Conclusion and future directions

3. Introduction
Janus geometries are dual to interface (conformal) field theories, which are well-controlled deformations of the AdS/CFT correspondence D. Bak, M. Gutperle and S. Hirano, JHEP 0305, 072 (2003) [hep- th/ ]; JHEP 0702, 068 (2007) [hep-th/ ] - A. B. Clark, D. Z. Freedman, A. Karch and M. Schnabl, Phys. Rev. D 71, (2005) [hep-th/ ]; E. D'Hoker, J. Estes and M. Gutperle, JHEP 0706, 022 (2007) [arXiv: [hep-th]].

4. Short summary of AdS/CFT to explain Janus field theory.
The relation between the bulk field and boundary quantities.
for example scalar field ; Asymptotic behavior;
These two functions are interpreted as a source and the Expectation value of corresponding dual operator.
More explicitly, the partition function of the field theory can be written in terms of a gravity action.

5. Massless Scalar If the bulk scalar field is massless,
Then, the field value at the boundary (z=0) corresponds to a dimensionless source.
We may consider cases where this source is a dimensionless coupling constant.
When the operator is the Lagrangian operator of SYM theory, the asymptotic behavior of the scalar field is known as
Thus boundary value of the scalar field corresponds to coupling constant.

6. This is the Lagrangian of the Janus field theory.
The coupling constant is discrete and the deformation induces the interface Lagrangian.
By AdS/CFT, there is a dual geometry to this field theory.
The geometry is an asymptotically AdS space deformed by a massless scalar( Dilaton ). - D. Bak, M. Gutperle and S. Hirano, JHEP 0305, 072 (2003) [hep- th/ ]; JHEP 0702, 068 (2007) [hep-th/ ]

7. If the field theory system has a finite temperature
Corresponding bulk geometry is a black hole with the scalar hair.

8. A black Janus geometry is dual to the finite temperature version of the interface (conformal) field theory.
An exact solution for the 3D black Janus geometry( dual to 2 dimensional field theory) is available in - D. Bak, M. Gutperle and R. A. Janik, JHEP 1110, 056 (2011) [arXiv: [hep-th]], - D. Bak and H. Min, JHEP 1403, 046 (2014) [arXiv: [hep- th]]. (Perturbative Analysis)
The dual geometry is given by a black brane with a discrete scalar hair.

9. Exact solution
Numerical solution

10. Now we want to extend the numerical study for more general cases.
One difficulty in these trials is the discrete boundary conditions on the scalar field which generates a quite big numerical error.
To mitigate the error, we selected a Monte Carlo method.
The MC method is independent of the boundary conditions(Advantage) and it probes the geometry by random walks.(Another useful insight ?)
We reconsider it as a geometric interpretation of Monte Carlo method.
We call this by “geometric MC method”.

11. As a first step, we only consider Janus deformation from a scalar field without back reaction
The system we have to solve is
The background is a black brane solution.
The scalar field equation is

12. The condition of the scalar field at the boundary of AdS space is
At the horizon we will impose the regularity condition for the smooth configuration of the scalar field.
The condition is a little bit complicated form.
Fortunately, the 3 dimensional scalar system admits an exact solution
We will consider this problem with the Monte Carlo method.

13. 3D Black Janus by Monte Carlo
Let us consider a simplest ODE case to get the idea of the MC method
Discretize the space
The derivatives in terms of the differences
The original equation becomes
One may notice that the coefficients look like probabilities. This is a general structure of this kind of differential equation.

14. Generating random number for each step
For given Dirichlet boundary condition, the numerical solution is given by

15. Coming back to the 3d case
partial differential equation
We may take differences of the scalar field as follows.

16. Then the differential equation takes the following form
This probabilities are given by
If you want to consider other equation, we can use same code with changing the probability and boundary conditions.

17. The rule for random walks.

18. The final numerical solution
For the smooth boundary condition on the horizon, we use the following approximation
This is equivalent to the regularity condition.

19. Define the relative error for the 3d case.
The numerical solution and the relative error.

20. 5 dimensional case (No analytic solution)
Since we check the power of the method, we move to 5d case - Planar interface
- Bag-like interface

21. This configuration is reminiscent of the MIT bag model for QCD.
To solve this, we may take an appropriate coordinate transformation and redefinition of the scalar field.
The equation becomes
The underlying scaling symmetry is broken leading to a potentially richer dynamics which may depend on the parameter RT.

22. 5D black janus
Black Janus with planar interface, the black brane give us the probabilities.

23. The numerical solution and the horizon profile of the scalar field

24. Bag-like model
Solutions and horizon profiles

25. Wilson line operator in this background
In the case with heavy quark approximation
The corresponding bulk physics is given by world volume action of a string attached to the boundary of AdS space.
The crucial part for us is that the effective string tension is proportional to
so that the region of smaller effective tension is dynamically preferred.
Thus in our bag-like model, the quark-antiquark pair inside the bag is dynamically preferred which is in accordance with the main idea of the MIT-bag model.

26. Discussion and Future Directions
By comparing our numerical result with the known exact solution in three dimensions, we found a good agreement confirming validity of our geometric MC method.
In five dimensions, numerical solutions with two different Janus boundary conditions (one with a planar shape and the other with a spherical shape) have been obtained.
the quark-anti-quark pair is dynamically preferred inside the bag as in the original MIT-bag model.
This method is appropriate for parallel computing. : Time and computing power is proportional to accuracy.
Our method may be applied to nonlinear problems including full back reaction of the gravity sector.
Once the back reaction is included, one may study various quantitative aspects of our 5D system including spectrum of mesons and their melting at finite temperature.
This could give us a new interpretation of geometry in terms of probabilities which could shed light on understanding quantum gravity.

27. Thank you very much !